Schreiber
integration of ∞-Lie algebroid valued differential forms

An integration of ∞-Lie algebroid valued differential forms $\omega :{\Pi }^{\inf }(X)\to 𝔞$ is an extension $\int \omega$ from the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$ to the finite path ∞-groupoid $\Pi (X)$

$$\begin{array}{ccc}{\Pi }^{\inf }(X)& \stackrel{\omega }{\to }& 𝔞\\ \downarrow & & \downarrow \\ \Pi (x)& \stackrel{\int \omega }{\to }& A\end{array}$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(x) &\stackrel{\int \omega}{\to}& A }

after injecting the ∞-Lie algebroid $𝔞$ into one of its corresponding ∞-Lie groupoids $A$.

Idea

Assume a context given by a smooth (∞,1)-topos with its notion of ∞-Lie theory.

For $A$ some ∞-Lie groupoid and $U\in C$ a test domain, a morphism $\varphi :{\Pi }^{\inf }(U)\to A$ from the infinitesimal path ∞-groupoid factors – by definition of ∞-Lie algebroid – through the ∞-Lie algebroid $𝔞$ of $A$

$${\Pi }^{\inf }(U)\stackrel{\omega }{\to }𝔞\hookrightarrow A$$\Pi^{inf}(U) \stackrel{\omega}{\to} \mathfrak{a} \hookrightarrow A

The first morphism here constitutes the set of ∞-Lie algebroid valued differential forms on $U$ that is encoded by $\varphi$.

We want to specify what it means to integrate these forms.

Whereas $\omega$ colors infinitesimal intervals on $U$ by infinitesimal morphisms in $A$, its integration should be a rule that labels finite intervals by finite morphisms, such that the restriction to any infinitesimal interval within a finite interval reproduces the assignment of $omgea$.

Formally, this desideratum clearly means that integration $\int \omega$ of $\omega$ is an extension

$$\begin{array}{ccc}{\Pi }^{\inf }(U)& \stackrel{\omega }{\to }& A\\ \downarrow & {\nearrow }_{\int \omega }\\ \Pi (U)\end{array}$$\array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow & \nearrow_{\int \omega} \\ \Pi(U) }

of $\omega$ through the inclusion ${\Pi }^{\inf }(U)\hookrightarrow \Pi (U)$ of the infinitesimal path ∞-groupoid in the path ∞-groupoid.

More generally, we can ask for a relative integration : we may have a situation where a coloring of finite intervals

$$\Pi (U)\to B$$\Pi(U) \to B

is already given, but infinitesimally there is also a lift specified of this through some map $A\to B$, i.e. a diagram

$$\begin{array}{ccc}{\Pi }^{\inf }(U)& \stackrel{\omega }{\to }& A\\ \downarrow & & \downarrow \\ \Pi (U)& \to & B\end{array}$$\array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow && \downarrow \\ \Pi(U) &\to& B }

Integration should exist when the map $A\to B$ in principle admits such a lift and should be given by that lift. It should be unique up to equivalence.

Details

We discuss a case in which integration of $\mathrm{\infty }$-Lie algebroid valued forms always exsist.

Let $𝔞$ be an ∞-Lie algebroid.

Write for short ${𝔞}_{\#}:={𝔞}_{{\#}^{\inf }}$ in the following for the image of $𝔞$ under the right adjoint of the infinitesimal path ∞-groupoid functor ${\Pi }^{\inf }$. And write ${𝔞}_{\#0}$ for the corresponding presheaf of 0-cells.

By the reasoning an ∞-Lie differentiation and integration we have

• the $\mathrm{\infty }$-Lie algebroid of $𝔞$ is $𝔞$ itself:

  $$\mathrm{Lie}(𝔞):={\Pi }^{\inf }({𝔞}_{\#0})=𝔞.$$Lie(\mathfrak{a}) := \Pi^{inf}(\mathfrak{a}_{# 0}) = \mathfrak{a} \,.

• the ∞-Lie groupoid integrating $𝔞$ is

  $$A=\Pi ({𝔞}_{\#0}).$$A = \Pi(\mathfrak{a}_{# 0}) \,.

For $X$ simplicially discrete, every morphism

$$\omega :{\Pi }^{\inf }(X)\to 𝔞$$\omega : \Pi^{inf}(X) \to \mathfrak{a}

corresponding by adjunction to a morphism

$$\tilde{\omega }:X\to {𝔞}_{{\#}_0}$$\tilde \omega : X \to \mathfrak{a}_{#_0}

induces a morphism

$$\int \omega :=\Pi (\tilde{\omega }):\Pi (X)\to \Pi ({𝔞}_{\#0})=A$$\int \omega := \Pi(\tilde \omega) : \Pi(X) \to \Pi(\mathfrak{a}_{# 0}) = A

that fits into the diagram

$$\begin{array}{ccccc}{\Pi }^{\inf }(X)& \stackrel{\omega }{\to }& {\Pi }^{\inf }({𝔞}_{\#0})& =& 𝔞\\ \downarrow & & \downarrow & & \downarrow \\ \Pi (X)& \stackrel{\int \omega }{\to }& \Pi ({𝔞}_{\#0})& =& A\end{array}.$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \Pi^{inf}(\mathfrak{a}_{# 0}) &=& \mathfrak{a} \\ \downarrow && \downarrow && \downarrow \\ \Pi(X) &\stackrel{\int \omega}{\to}& \Pi(\mathfrak{a}_{# 0}) & = & A } \,.